A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $\mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of indecomposable representations or as the direct sum of cyclic submodules generated by homogeneous elements. An interval basis for $M$ is a set of homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$ generated by them is direct and equal to $M$. We introduce a novel algorithm to compute an interval basis for $M$. Based on a flag of kernels of the structure maps, our algorithm is suitable for parallel or distributed computation and does not rely on a presentation of $M$. This algorithm outperforms the approach via the presentation matrix and Smith Normal Form. We specialize our parallel approach to persistent homology modules, and we close by applying the proposed algorithm to tracking harmonics via Hodge decomposition.

翻译：设系数域为$\mathbb{F}$的持续模$M$是$A_n$型等方向箭图的有限维线性表示，等价于多项式环$\mathbb{F}[x]$上的分次模。众所周知，$M$可分解为不可分解表示直和，或由齐次元素生成的循环子模直和。$M$的区间基是一组齐次元素集合，使得由它们生成的循环子模的直和恰好等于$M$。本文提出一种计算$M$区间基的新算法。该算法基于结构映射核的旗标，适用于并行或分布式计算，且无需依赖$M$的表示形式。相较于通过表示矩阵与Smith标准形的传统方法，本算法在性能上具有优势。我们将该并行方法特化应用于持续同调模，最后通过Hodge分解追踪谐波的应用实例验证所提算法。